# Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theorems on the Solution of Euler’s Differential Equation

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

**Example**

**1.**

**Remark**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**2.**

**Theorem**

**3.**

## 3. Preliminaries on Distribution Theory

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

#### 3.1. Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Condition**

**1.**

**Lemma**

**8.**

**Proof.**

#### 3.2. Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$, as well as in the Space ${\mathcal{D}}^{\prime}$

**Remark**

**4.**

**Remark**

**5.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

#### 3.3. Regular Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Lemma**

**14.**

**Lemma**

**15.**

## 4. Euler’s Equation in the Space of Distributions ${\mathcal{D}}_{R}^{\prime}$

**Remark**

**6.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Remark**

**7.**

**Lemma**

**18.**

**Proof.**

**Theorem**

**4.**

**Remark**

**8.**

**Remark**

**9.**

**Proof**

**of**

**Theorem**

**4.**

**Example**

**3.**

**Remark**

**10.**

**Example**

**4.**

**Theorem**

**5.**

**Proof.**

**Remark**

**11.**

**Example**

**5.**

**Theorem**

**6.**

**Remark**

**12.**

**Example**

**6.**

**Example**

**7.**

## 5. Euler’s Equation Studied in Nonstandard Analysis

**Theorem**

**7.**

**Proof.**

#### 5.1. AC-Laplace Transform of Euler’s Equation

**Theorem**

**8.**

**Remark**

**13.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Solution of a Linear Differential Equation with Constant Coefficients

**Theorem**

**A1.**

## References

- Morita, T.; Sato, K. A study on the Solution of Linear Differential Equations with Polynomial Coefficients. J. Adv. Math. Comput. Sci.
**2018**, 28, 1–15. [Google Scholar] [CrossRef] - Morita, T.; Sato, K. On the Solution of Linear Differential Equations with Polynomial Coefficients near the Origin and Infinity. J. Adv. Math. Comput. Sci.
**2018**, 29, 1–17. [Google Scholar] [CrossRef] - Ince, E.L. Ordinary Differential Equations; Dover Publ. Inc.: New York, NY, USA, 1956. [Google Scholar]
- Poole, E.G.C. Introduction to the Theory of Linear Differential Equations; Oxford U. P.: London, UK, 1936. [Google Scholar]
- Sangsuwan, A.; Nonlaopon, K.; Orakitjaroen, S. The Generalized Solutions of a Certain nth Order Cauchy-Euler Equation. Asian-Eur. J. Math.
**2020**, 13, 2050047. [Google Scholar] [CrossRef] - Jhanthanam, S.; Nonlaopon, K.; Orakitjaroen, S. Generalized Solutions of the Third-order Cauchy-Euler Equation in the Space of Right-sided Distributions via Laplace Transform. Mathematics
**2019**, 7, 376. [Google Scholar] [CrossRef] [Green Version] - Sangsuwan, A.; Nonlaopon, K.; Orakitjaroen, S.; Mirumbe, I. The Generalized Solutions of the nth Order Cauchy-Euler Equation. Mathematics
**2019**, 7, 932. [Google Scholar] [CrossRef] [Green Version] - Nishimoto, K. An Essence of Nishimoto’s Fractional Calculus; Descartes Press: Koriyama, Japan, 1991. [Google Scholar]
- Nishimoto, K. Kummer’s Twenty-four Functions and N-Fractional Calculus. Nolinear Anal. Theory Methods Appl.
**1997**, 30, 1271–1282. [Google Scholar] [CrossRef] - Morita, T.; Sato, K. Asymptotic Expansions of Fractional Derivatives and Their Applications. Mathematics
**2015**, 3, 171–189. [Google Scholar] [CrossRef] [Green Version] - Morita, T.; Sato, K. Kummer’s 24 Solutions of the Hypergeometric Differential Equations with the Aid of Fractional Calculus. Adv. Pure Math.
**2016**, 6, 180–191. [Google Scholar] [CrossRef] [Green Version] - Sneddon, I.N. Special Functions of Mathematical Physics and Chemistry; Longman Inc.: New York, NY, USA, 1980. [Google Scholar]
- Nishimoto, K. Applications to the Solutions of Linear Second Order Differential Equations of Fuchs Type. In Fractional Calculus; McBride, A.C., Roach, G.F., Eds.; Pitman Advanced Publishing Program: Boston, MA, USA, 1985. [Google Scholar]
- Schwartz, L. Théorie des Distributions; Hermann: Paris, France, 1966. [Google Scholar]
- Gelfand, I.M.; Silov, G.E. Generalized Functions; Academic Press Inc.: New York, NY, USA, 1964; Volume 1. [Google Scholar]
- Vladimirov, V.S. Methods of the Theory of Generalized Functions; Taylor & Francis Inc.: New York, NY, USA, 2002. [Google Scholar]
- Zemanian, A.H. Distribution Theory and Transform Analysis; Dover Publ. Inc.: New York, NY, USA, 1965. [Google Scholar]
- Morita, T.; Sato, K. Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform. Appl. Math.
**2014**, 5, 1209–1219. [Google Scholar] [CrossRef] [Green Version] - Morita, T.; Sato, K. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. Mathematics
**2016**, 4, 19. [Google Scholar] [CrossRef] [Green Version] - Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis; Cambridge U.P.: Cambridge, UK, 1935. [Google Scholar]
- Robert, A.M. Nonstandard Analysis; Dover Publ. Inc.: New York, NY, USA, 2003. [Google Scholar]
- Ghil, B.; Kim, H. The Solution of Euler-Cauchy Equation Using Laplace Transform. Int. J. Math. Anal.
**2015**, 9, 2611–2618. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Morita, T.; Sato, K.-i.
Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus. *Mathematics* **2020**, *8*, 2117.
https://doi.org/10.3390/math8122117

**AMA Style**

Morita T, Sato K-i.
Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus. *Mathematics*. 2020; 8(12):2117.
https://doi.org/10.3390/math8122117

**Chicago/Turabian Style**

Morita, Tohru, and Ken-ichi Sato.
2020. "Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus" *Mathematics* 8, no. 12: 2117.
https://doi.org/10.3390/math8122117